Embedded newform 80.12.a.b.1.1

• Label: 80.12.a.b.1.1
• Level: $80$
• Weight: $12$
• Character: 80.1
• Self dual: yes
• Analytic conductor: $61.467$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(61.4674544448\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	80.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-520.000 q^{3} +3125.00 q^{5} -15148.0 q^{7} +93253.0 q^{9} -369324. q^{11} +877926. q^{13} -1.62500e6 q^{15} +3.28871e6 q^{17} -1.56080e6 q^{19} +7.87696e6 q^{21} +1.28994e7 q^{23} +9.76562e6 q^{25} +4.36249e7 q^{27} +4.63225e7 q^{29} +5.94275e7 q^{31} +1.92048e8 q^{33} -4.73375e7 q^{35} +6.66072e8 q^{37} -4.56522e8 q^{39} +6.74573e8 q^{41} -1.18838e9 q^{43} +2.91416e8 q^{45} -2.26148e9 q^{47} -1.74786e9 q^{49} -1.71013e9 q^{51} -3.09924e9 q^{53} -1.15414e9 q^{55} +8.11614e8 q^{57} +1.38382e9 q^{59} -1.23091e9 q^{61} -1.41260e9 q^{63} +2.74352e9 q^{65} -1.75593e10 q^{67} -6.70767e9 q^{69} +2.78727e9 q^{71} -2.12373e10 q^{73} -5.07812e9 q^{75} +5.59452e9 q^{77} +4.30659e10 q^{79} -3.92044e10 q^{81} +2.15524e10 q^{83} +1.02772e10 q^{85} -2.40877e10 q^{87} -9.21446e10 q^{89} -1.32988e10 q^{91} -3.09023e10 q^{93} -4.87749e9 q^{95} -8.04846e10 q^{97} -3.44406e10 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	−520.000	−1.23548	−0.617741	−	0.786382i	\(-0.711952\pi\)
−0.617741	+	0.786382i				\(0.711952\pi\)
\(5\)	3125.00	0.447214
			\(7\)	−15148.0	−0.340656	−0.170328	−	0.985387i	\(-0.554483\pi\)
−0.170328	+	0.985387i				\(0.554483\pi\)
\(11\)	−369324.	−0.691429	−0.345715	−	0.938340i	\(-0.612363\pi\)
			−0.345715	+	0.938340i	\(0.612363\pi\)
\(13\)	877926.	0.655797	0.327899	−	0.944713i	\(-0.393660\pi\)
			0.327899	+	0.944713i	\(0.393660\pi\)
\(17\)	3.28871e6	0.561767	0.280883	−	0.959742i	\(-0.409373\pi\)
			0.280883	+	0.959742i	\(0.409373\pi\)
\(19\)	−1.56080e6	−0.144611	−0.0723055	−	0.997383i	\(-0.523036\pi\)
			−0.0723055	+	0.997383i	\(0.523036\pi\)
\(23\)	1.28994e7	0.417893	0.208947	−	0.977927i	\(-0.432996\pi\)
			0.208947	+	0.977927i	\(0.432996\pi\)
\(29\)	4.63225e7	0.419375	0.209688	−	0.977768i	\(-0.432755\pi\)
			0.209688	+	0.977768i	\(0.432755\pi\)
\(31\)	5.94275e7	0.372819	0.186410	−	0.982472i	\(-0.440315\pi\)
			0.186410	+	0.982472i	\(0.440315\pi\)
\(37\)	6.66072e8	1.57911	0.789554	−	0.613681i	\(-0.210312\pi\)
			0.789554	+	0.613681i	\(0.210312\pi\)
\(41\)	6.74573e8	0.909322	0.454661	−	0.890664i	\(-0.349760\pi\)
			0.454661	+	0.890664i	\(0.349760\pi\)
\(43\)	−1.18838e9	−1.23276	−0.616381	−	0.787448i	\(-0.711402\pi\)
			−0.616381	+	0.787448i	\(0.711402\pi\)
\(47\)	−2.26148e9	−1.43832	−0.719159	−	0.694846i	\(-0.755473\pi\)
			−0.719159	+	0.694846i	\(0.755473\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.12.a.b.1.1		1
4.3	odd	2		40.12.a.a.1.1	✓	1
20.3	even	4		200.12.c.a.49.2		2
20.7	even	4		200.12.c.a.49.1		2
20.19	odd	2		200.12.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.12.a.a.1.1	✓	1	4.3	odd	2
80.12.a.b.1.1		1	1.1	even	1	trivial
200.12.a.a.1.1		1	20.19	odd	2
200.12.c.a.49.1		2	20.7	even	4
200.12.c.a.49.2		2	20.3	even	4